288 X EXISTENCE THEOREMS

(10.5.5) Let f be continuously differentiate in I x H ifK = C, locally lip-
schitzian in I x H z/K = R. Suppose v is a solution of (10.4.1) defined in an
open ball J : \t - t0 \ < r, such that ] c I, that v(S) c= H, and that t -»/(/, v(t)) is
bounded in J. 77zŁ/z there exists a ball J' : |f — /0| < r' contained in I, w///7 r' > r,
solution of (1 0.4.1 ) defined in J' #/?<af coinciding with v in J.

(a) K = R. By assumption, we have \\f(t, 0(0) II ^ M for t e J, hence
||z/(0li < M in the complement of an at most denumerable subset of J. This
implies \\v(s) — v(t)\\ < M\s - t\ for s, t in J by the mean value theorem
(8.5.2). From the Cauchy convergence criterion (3.14.6) we conclude that
the limits v((t0 - r)+) and v((tQ + r)-) exist and belong to v(J) c H. By
(10.4.6), there exists a solution wi (resp. w2) of x' = f(t, x) defined in an open
ball Ui (resp. U2) of center tQ 4- /-(resp. t0 — r), contained in I, and taking the
value v((t0 + r)-) (resp. v((t0 - r)+)) at this point; from (10.5.4) it follows
in addition that vt^ (resp. w2) coincides with v in Uj n J (resp. U2 n J), and the
proof is therefore concluded in that case. (One may observe that it has not
been necessary to check the existence of derivatives on the left or on the
right for v (extended by continuity) nor for v^ and w2 , at the points /0 — r
and t0 + r.)

(b) K = C. For any complex number C such that |CI = 1, put
t = /0 + C$v with s ^ 0, and v^(s) = v(t0 + Łs). Then the same argument as in
(a) proves that v^r— ) exists and is in H; hence there exists a solution w^ of
x' =/(/, x) defined in an open ball Vc of center t0 + Ł/» contained in J, such
that w^(t0 + CO = i\(r— ). From (10.5.4) it follows that w$ and v coincide in
the intersection of J n V^ with the segment of extremities t0 and /0 -f Łr; as
these functions are analytic in J n Vc, they coincide in J n Vc by (9.4.4).
Now cover the compact set \t — t0\ = r with finitely many balls VSl (1 ^ / < m);
if VCi n Vc/ ^ 0, the functions wCi and w^ coincide in V?. n V^ , for both
coincide with v in the nonempty open set J n V^ n V^ , and we have only to
apply (9.4.2) (to show that the preceding intersection is not empty, remark that
the assumption implies /*|C,- — C/l <Pi + Pj, where pt, PJ are the radii of
V^ and VCj; hence there is Ae]0, 1[ such that rA|(; — C^l < Pi

and r(l - A)|C, - Cyl < P^; it follows that the point t0 + r((l - A)Cf +
belongs to J n Vc. n V^.). There is therefore a solution of xf =/(/, ^c) equal to
v in J, to vv^ in each of the V^ , and there is an open ball of center /0 and radius
r' > r contained in the union of these sets (3.17.11), which ends the proof.

(10.5.5.1) It follows from (10.5.5) that if r0 is the l.u.b. of all numbers r such
that J c I and v(J) c H, either r0 = 4- oo, or, if J0 is the open ball \t — t0 \ < r0,
one of the two relations J0 Ł I, z?(J0) ^ H holds.d suppose a is an essential singularity of /(Section 9.1 5). Show that C — /(V) is empty